All-frequency PMD compensator and emulator

ABSTRACT

A PMD device includes a first stage that receives a signal and performs rotation about {1,0,0} on the signal. The first stage outputs a first signal. A second stage receives the first signal and performs a rotation about {0,0,1 } on the first signal. The second stage outputs a second signal that represents the alignment the PMD of various frequencies into a common direction. A third stage receives the second signal and provides the necessary frequency dependent variable in the {1,0,0} direction to cancel the PMD in any specified frequency range.

PRIORITY INFORMATION

This application claims priority from provisional application Ser. No.60/491,711 filed Aug. 1, 2003, which is incorporated herein by referencein its entirety.

BACKGROUND OF THE INVENTION

The invention relates to the field polarization mode dispersion, and inparticular to a device that can compensate and emulate within a largefrequency range.

As the bit rate of a single channel increases, a compensator thatcancels first order polarization mode dispersion (PMD) is no longersufficient since higher order PMD dominates the signal's degradation.Typical higher order PMD compensators are multistage devices that havemany degrees of freedom. If the compensation scheme relies on feedbackloops, the complexity involved in searching the optimum of manyparameters may hinder the development of a fast and stable compensator.Thus, feed-forward approach to PMD compensation appears more feasiblewhen higher order PMD compensation need to be addressed. Recently,several schemes of real-time PMD estimation technique had been proposed.

Previous approaches to all-frequency PMD compensation aim to generatethe inverse of transmission fiber's Jones matrix U(ω) for allfrequencies. Most of these efforts concentrate on the synthesis offilter response to approximate this unitary matrix, U(ω). In onearchitecture, it describes using all-pass filters to invert the measuredJones matrix U(ω) approximately. The approach is very promising as itcan be compactly integrated using planar waveguides.

SUMMARY OF THE INVENTION

The invention provides an architecture that suppress the informationabout the isotropic dispersion in the Stokes space formulation, and showthat the knowledge of the output PMD vector, {right arrow over(τ)}_(f)(ω), as a function of frequency is sufficient for theconstruction of an all-frequency PMD compensator. The Jones matrix U(ω)varies more rapidly than its frequency dependent counterpart,jU_(ω)(ω)U⁺(ω), which represents the output PMD in Jones Spaceformulation. Thus, one of the main advantages of compensating the outputPMD {right arrow over (τ)}_(f)(ω) is that the real-time monitoring andcompensation can be performed at a slower rate than compensating theJones matrix. In addition, monitoring of the fiber's output PMD {rightarrow over (τ)}_(f)(ω) does not require any knowledge of the input stateof polarization (SOP), thus polarization scrambling of the input SOP canbe used to improve estimation accuracy of the PMD monitoring.

According to one aspect of the invention, there is provided a PMDdevice. The PMD device includes a first stage that receives a signal andperforms rotation about {1,0,0} on the signal. The first stage outputs afirst signal. A second stage receives the first signal and performs arotation about {0,0,1} on the first signal. The second stage outputs asecond signal that represents the alignment the PMD of variousfrequencies into a common direction. A third stage receives the secondsignal and provides the necessary frequency dependent variableDifferential Group Delay (DGD) in the {1,0,0} direction to cancel thePMD in any specified frequency range.

According to another aspect of the invention, there is provided a PMDcompensator. The PMD compensator includes a first stage that receives asignal and performs rotation about {1,0,0} on the signal. The firststage outputs a first signal. A second stage receives the first signaland performs a rotation about {0,0,1} on the first signal. The secondstage outputs a second signal that represents the alignment the PMD ofvarious frequencies into a common direction. A third stage receives thesecond signal and provides the necessary frequency dependent variableDifferential Group Delay (DGD) in the {1,0,0} direction to cancel thePMD in any specified frequency range.

According to another aspect of the invention, there is provided a PMDcompensator. The PMD compensator includes a first stage that receives asignal and performs rotation about {1,0,0} on the signal. The firststage outputs a first signal. A second stage receives the first signaland performs a rotation about {0,0,1} on the first signal. The secondstage outputs a second signal that represents the alignment the PMD ofvarious frequencies into a common direction. A third stage receives thesecond signal and provides the necessary frequency dependent variableDGD in the {1,0,0} direction to cancel the PMD in any specifiedfrequency range.

According to another aspect of the invention, there is provided a methodof operating a PMD device. The method includes receiving at a firststage a signal and performing rotation about {1,0,0} on the signal. Thefirst stage outputs a first signal. The method includes receiving at asecond stage the first signal and performing a rotation about {0,0,1} onthe first signal. The second stage outputs a second signal thatrepresents the alignment the PMD of various frequencies into a commondirection. Furthermore, the method includes receiving at a third stagethe second signal and providing the necessary frequency dependentvariable DGD in the {1,0,0} direction to cancel the PMD in any specifiedfrequency range.

BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1A-1B are schematic diagrams illustrating a PMD compensator inaccordance with the invention;

FIG. 2 is a schematic diagram illustrating a detailed depiction of thePMD compensator of the invention; and

FIGS. 3A-3D are graphs demonstrating the required rotation angles forvarious stages of the PMD compensator as a function of frequency;

FIGS. 4A-4D are graphs demonstrating eye-diagrams of the signals shownin FIGS. 3A-3D;

FIG. 5A is a graph illustrating the eye-diagrams of the output signalsfrom the fiber before compensation; FIG. 5B is a graph illustrating theeye-diagrams of the signals after the PMD compensation using exactrotation angles; FIG. 5C is a graph demonstrating an eye-diagram of thesignals after PMD compensation using approximated rotation anglessynthesized by all-pass filter;

FIGS. 6A-6B are schematic diagrams illustrating another exemplaryembodiment of a PMD compensator in accordance with the invention;

FIG. 7 is a graph demonstrating the rotational angles for a simulatedfiber; and

FIG. 8 is a graph illustrating an output signal from the fiber afteremulation.

DETAILED DESCRIPTION OF THE INVENTION

The invention describes an architecture of a compensator that cancompensate PMD for all frequency based solely on the knowledge of theoutput PMD vectors, {right arrow over (τ)}_(f)(ω), as a function offrequency. It employs four stages. The first two stages provide anequivalent frequency dependent polarization rotation effect to align allPMD vectors into a common direction, and the third stage compensates thefrequency dependent variable DGD. The last stage basically compensatesthe isotropic dispersion introduced by the first three stages. Thesestages can be implemented using all-pass filters (APF), however, othersimilar components that generate frequency-dependent phase profiles canbe used. The required phase responses of these filters are distinctlydifferent from other approaches in the prior art to approximate theinverse Jones matrix while the aim is to compensate itsfrequency-dependent part (i.e. jU_(ω)(ω)U⁺(ω)). In Stokes spaceformulation, the synthesis algorithm of the required rotation angles ofthe various stages using PMD concatenation rules is presented, and thesefilter responses are shown to be realistic and can be approximated byrelatively few APF. Through simulations, significant improvement of thesignal quality using such architecture is demonstrated.

For simplicity, one may neglect polarization-dependent losses, andassume that using the real-time PMD monitoring scheme, the fiber'soutput PMD vector, {right arrow over (τ)}_(f)(ω) is known, as a functionof frequency. Expressing the PMD as a Taylor expansion about the centerfrequency, ω_(o), is avoided since this introduces complicatedhigher-order PMD terms such as$\frac{\mathbb{d}\overset{->}{\tau}}{\mathbb{d}\omega},\frac{\mathbb{d}^{2}\overset{->}{\tau}}{\mathbb{d}\omega^{2}}$etc. Instead, the whole PMD spectrum is treated as first order PMDvectors that vary from frequency to frequency. For this reason, one canchoose to call the proposed scheme “All-Frequency”, instead of“All-Order”, PMD compensator. shaping schemes using a spatial lightmodulator or using a deformable mirror. For stage 1 and 3, the light ispolarization beam split into horizontal and vertical polarization. Onepolarization is then dispersed spatially by a diffraction grating toform a line spectrum across the deformable mirror or spatial lightmodulator. The mirror (or modulator) is programmed to produce thedesired changes in the spectral phase across the bandwidth. When the twopolarizations are combined using a polarization beam combiner, thiscreates frequency-dependent rotation about {1,0,0} in Stokes space.Stage 2 has the same setup as stage 1 and 3 except for the additionalquarter-wave plates before and after the setup. This makes stage 2 anequivalent rotation about {0,0,1} in Stokes space. Thus, by programmingappropriate spectral phase, the required rotation angles θ₁(ω), θ₂(ω)and θ₃(ω) can be produced as calculated from equation (6-9) and 12-13).

The third promising implementation is based on all-pass Filters (APFs)integrated on a planar lightwave circuit as shown in FIG. 6. An all-passfilter (APF) has unity magnitude response; and, by cascading them, onecan engineer the phase response to approximate any desired response.This approach is common in the field of electrical circuit design anddigital signal processing and has been used extensively for phaseequalization in these fields.

Two common implementations of optical APF are (a ring resonator coupledto a straight waveguide and b) an etalon that has a perfect mirror onone side. The former is preferred for compact integrated optics. Thelatter is also known as a Gires-Toumois interferometer. The powercoupling coefficient into the resonator and the resonant frequency arethe two main filter parameters to adjust when engineering the phaseresponse. It has been demonstrated had demonstrated that the tuning ofthese two filter

FIG. 1A shows the schematic of the proposed architecture. A portion ofthe signal is tapped at the fiber output for real-time PMD monitoring.Based on the knowledge of the PMD vector data, one can control the4-stage compensator. At each angular frequency ω (ω=2πf where f is theoptical frequency), the fiber's output PMD vector, {right arrow over(τ)}_(f)(ω), has three parameters $\begin{pmatrix}{\tau_{fx}(\omega)} \\{\tau_{fy}(\omega)} \\{\tau_{fz}(\omega)}\end{pmatrix}.$The first three stages of the compensator provide the three degrees offreedom at each frequency for the PMD cancellation. Stage 4 only dealswith the isotropic dispersion introduced by the first three stages. InStokes space, Stage 1 is a rotation about {1,0,0} with rotation angleθ₁(ω) as a function of frequency while Stage 2 is a rotation about the{0,0,1} with rotation angle θ₂(ω) which is also a function of frequency.The combined effect of stage 1 and 2 is equivalent to that of afrequency-dependent polarization controller that aligns the PMD ofvarious frequencies into a common direction, in this case, one canchoose it to be {1,0,0}.

FIG. 1B illustrates this rotation effect. After aligning the PMD vectors40, 42, Stage 3 provides the necessary frequency dependent variable DGD${\tau_{3}(\omega)} = \frac{\mathbb{d}{\theta_{3}(\omega)}}{\mathbb{d}\omega}$in the {1,0,0} direction to cancel the PMD. Since the rotation angles ofStage 1 and 2 are a function of frequency, they possess substantial DGDthat cannot be neglected. This is in contrary to polarizationcontrollers that employ low-order wave-plates. Finally, Stage 4compensates for the isotropic dispersion introduced by the first threestages due to imperfect fitting of the rotation angles. This will befurther discussed. In Stokes space, the rotation matrix of Stage 1 is$\begin{matrix}{{R_{1}(\omega)} = \begin{pmatrix}1 & 0 & 0 \\0 & {\cos\quad{\theta_{1}(\omega)}} & {{- \sin}\quad{\theta_{1}(\omega)}} \\0 & {\sin\quad{\theta_{1}(\omega)}} & {\cos\quad{\theta_{1}(\omega)}}\end{pmatrix}} & (1)\end{matrix}$and the rotation matrix of Stage 2 is $\begin{matrix}{{R_{2}(\omega)} = \begin{pmatrix}{\cos\quad{\theta_{2}(\omega)}} & {{- \sin}\quad{\theta_{2}(\omega)}} & 0 \\{\sin\quad{\theta_{2}(\omega)}} & {\cos\quad{\theta_{2}(\omega)}} & 0 \\0 & 0 & 1\end{pmatrix}} & (2)\end{matrix}$

Since${\overset{->}{\tau} \times \frac{\mathbb{d}R}{\mathbb{d}\omega}R^{+}},$the corresponding PMD vectors of stage 1 and 2 are${{\overset{->}{\tau}}_{1}(\omega)} = {{\begin{pmatrix}\frac{\mathbb{d}{\theta_{1}(\omega)}}{\mathbb{d}\omega} \\0 \\0\end{pmatrix}\quad{and}\quad{{\overset{->}{\tau}}_{2}(\omega)}} = \begin{pmatrix}0 \\0 \\\frac{\mathbb{d}{\theta_{2}(\omega)}}{\mathbb{d}\omega}\end{pmatrix}}$respectively. One can assume the fiber's PMD,${{{\overset{->}{\tau}}_{f}(\omega)} = \begin{pmatrix}{\tau_{fx}(\omega)} \\{\tau_{fy}(\omega)} \\{\tau_{fz}(\omega)}\end{pmatrix}},$is known. To analyze the PMD after Stage 2, {right arrow over (Γ)}₂(ω),one can use the PMD concatenation rule. $\begin{matrix}\begin{matrix}{{{\overset{->}{\Gamma}}_{2}(\omega)} = {{{\overset{->}{\tau}}_{2}(\omega)} + {{R_{2}(\omega)}{{\overset{->}{\tau}}_{1}(\omega)}} + {{R_{2}(\omega)}{R_{1}(\omega)}{{\overset{->}{\tau}}_{f}(\omega)}}}} \\{= \begin{pmatrix}\begin{matrix}{{\cos\quad{\theta_{2}( {\frac{\mathbb{d}\theta_{1}}{\mathbb{d}\omega} + \tau_{fx}} )}} - {\sin\quad{\theta_{2}( {{\tau_{fy}\cos\quad\theta_{1}} - {\tau_{fz}\sin\quad\theta_{1}}} )}}} \\{{\sin\quad{\theta_{2}( {\frac{\mathbb{d}\theta_{1}}{\mathbb{d}\omega} + \tau_{fx}} )}} + {\cos\quad{\theta_{2}( {{\tau_{fy}\cos\quad\theta_{1}} - {\tau_{fz}\sin\quad\theta_{1}}} )}}}\end{matrix} \\{\frac{\mathbb{d}\theta_{2}}{\mathbb{d}\omega} + ( {{\tau_{fy}\sin\quad\theta_{1}} + {\tau_{fz}\cos\quad\theta_{1}}} )}\end{pmatrix}}\end{matrix} & (3)\end{matrix}$Since one wants the PMD vector of all frequency to be aligned to {1,0,0}after Stage 2, one needs $\begin{matrix}{{{\sin\quad{\theta_{2}( {\frac{\mathbb{d}\theta_{1}}{\mathbb{d}\omega} + {\tau_{fx}(\omega)}} )}} + {\cos\quad{\theta_{2}( {{{\tau_{fy}(\omega)}\cos\quad\theta_{1}} - {{\tau_{fz}(\omega)}\sin\quad\theta_{1}}} )}}} = {0\quad{and}}} & (4) \\{{\frac{\mathbb{d}\theta_{2}}{\mathbb{d}\omega} + ( {{{\tau_{fy}(\omega)}\sin\quad\theta_{1}} + {{\tau_{fz}(\omega)}\cos\quad\theta_{1}}} )} = 0} & (5)\end{matrix}$

Solutions of Eqs. 4 and 5 yield the required rotation angles θ₁(ω) andθ₂(ω). To solve these equations, the following synthesis algorithm isused. At an initial frequency, ω_(initial), one can determine therotation angles of Stage 1 and 2 to bring {right arrow over(τ)}_(f)(ω_(initial)) aligned to {1,0,0}. These θ₁(ω_(initial)) andθ₂(ω_(initial)) provide the starting points for the algorithm. Forsubsequent frequency, one finds the rotation angles in a step-wisemanner: $\begin{matrix}{{\theta_{1}( {\omega + {\Delta\quad\omega}} )} \approx {{\theta_{1}(\omega)} + {\frac{\mathbb{d}\theta_{1}}{\mathbb{d}\omega}(\omega)\Delta\quad\omega}}} & (6) \\{{{\theta_{2}( {\omega + {\Delta\quad\omega}} )} \approx {{\theta_{2}(\omega)} + {\frac{\mathbb{d}\theta_{2}}{\mathbb{d}\omega}(\omega)\Delta\quad\omega}}}{where}} & (7) \\{{\frac{\mathbb{d}\theta_{1}}{\mathbb{d}\omega}(\omega)\quad{and}\quad\frac{\mathbb{d}\theta_{2}}{\mathbb{d}\omega}(\omega)\quad{are}\quad{given}\quad{by}\quad{{Eqs}.\quad(4)}\quad{and}\quad(5)\quad{as}}{{\frac{\mathbb{d}\theta_{1}}{\mathbb{d}\omega}(\omega)} = {{\cot\quad{{\theta_{2}(\omega)}\lbrack {{{\tau_{fz}(\omega)}\sin\quad{\theta_{1}(\omega)}} - {{\tau_{fy}(\omega)}\cos\quad{\theta_{1}(\omega)}}} \rbrack}} - {\tau_{fx}(\omega)}}}{and}} & (8) \\{{\frac{\mathbb{d}\theta_{2}}{\mathbb{d}\omega}(\omega)} - \lbrack {{{\tau_{fy}(\omega)}\sin\quad{\theta_{1}(\omega)}} + {{\tau_{fz}(\omega)}\cos\quad{\theta_{1}(\omega)}}} \rbrack} & (9)\end{matrix}$

In this way, one can successively synthesize the rotation angles ofStage 1 and 2 to align all the PMD vectors into {1,0,0} and the PMDvector after Stage 2 becomes $\begin{matrix}{{{\overset{->}{\Gamma}}_{2}(\omega)} = \begin{pmatrix}{{\cos\quad{\theta_{2}( {\frac{\mathbb{d}\theta_{1}}{\mathbb{d}\omega} + \tau_{fx}} )}} - {\sin\quad{\theta_{2}( {{\tau_{fy}\cos\quad\theta_{1}} - {\tau_{fz}\sin\quad\theta_{1}}} )}}} \\0 \\0\end{pmatrix}} & (10)\end{matrix}$Since Stage 3 is a rotation about {1,0,0}, its rotation matrix has noeffect on {right arrow over (Γ)}₂(ω). The PMD vector of Stage 3 is${{\overset{->}{\tau}}_{3}(\omega)} = {\begin{pmatrix}\frac{\mathbb{d}{\theta_{3}(\omega)}}{\mathbb{d}\omega} \\0 \\0\end{pmatrix}.}$After passing through Stage 3, the resultant PMD vector {right arrowover (Γ)}₃(ω) becomes $\begin{matrix}{{{\overset{->}{\Gamma}}_{3}(\omega)} = {{{\overset{->}{\tau}}_{3}(\omega)} + {{R_{3}(\omega)}{{\overset{->}{\Gamma}}_{2}(\omega)}\quad(11)}}} \\{= {{{\overset{->}{\tau}}_{3}(\omega)} + {{\overset{->}{\Gamma}}_{2}(\omega)}}} \\{= \begin{pmatrix}{\frac{\mathbb{d}{\theta_{3}(\omega)}}{\mathbb{d}\omega} + {\cos\quad{\theta_{2}( {\frac{\mathbb{d}\theta_{1}}{\mathbb{d}\omega} + \tau_{fx}} )}} - {\sin\quad{\theta_{2}( {{\tau_{fy}\cos\quad\theta_{1}} - {\tau_{fz}\sin\quad\theta_{1}}} )}}} \\0 \\0\end{pmatrix}}\end{matrix}$

To have zero resultant PMD for all frequencies after Stage 3, {rightarrow over (Γ)}₃(ω) must be zero for all frequencies. Thus, one needs$\begin{matrix}{\frac{\mathbb{d}{\theta_{3}(\omega)}}{\mathbb{d}\omega} = {{{- \cos}\quad{{\theta_{2}(\omega)}\lbrack {{\frac{\mathbb{d}\theta_{1}}{\mathbb{d}\omega}(\omega)} + {\tau_{fx}(\omega)}} \rbrack}} + \quad{\sin\quad{{\theta_{2}(\omega)}\lbrack {{{\tau_{fy}(\omega)}\cos\quad{\theta_{1}(\omega)}} - {{\tau_{fz}(\omega)}\sin\quad{\theta_{1}(\omega)}}} \rbrack}}}} & (12)\end{matrix}$Together with θ₁(ω), θ₂(ω),$\frac{\mathbb{d}\theta_{1}}{\mathbb{d}\omega}(\omega)\quad{and}\quad\frac{\mathbb{d}\theta_{2}}{\mathbb{d}\omega}(\omega)$(known from equations (6-9)), θ₃(ω) is synthesized by arbitrary fixingθ₃(ω_(initial))=0 and subsequent frequency by $\begin{matrix}{{\theta_{3}( {\omega + {\Delta\quad\omega}} )} \approx {{\theta_{3}(\omega)} + {\frac{\mathbb{d}\theta_{3}}{\mathbb{d}\omega}(\omega)\Delta\quad\omega}}} & (13)\end{matrix}$

In summary, to the synthesis algorithm, using Eqs. (6-9) and (12-13),one can synthesize the required rotation angle of θ₁(ω), θ₂(ω) and θ₃(ω)for complete PMD compensation at all frequency.

There are at least three physical implementations of the proposed4-stage AFPMD compensator. The first two implementations can be adaptedfrom the femto-second pulse parameters is feasible using thermal heatersin a Mach-Zehnder interferometer configuration of APF. In general, thephase response of a set of N_(i) APF is given by: $\begin{matrix}{{\Phi_{i}(\omega)} = {{N_{i}( {\pi - {\omega\quad T}} )} - {\sum\limits_{k}^{N_{i}}( {\phi_{k} + {2{\tan^{- 1}( \frac{r_{k}{\sin( {{\omega\quad T} + \phi_{k}} )}}{1 - {r_{k}{\cos( {{\omega\quad T} + \phi_{k}} )}}} )}}} )}}} & (14)\end{matrix}$where φ_(k) determines the cavity's resonant frequency, r_(k) is thepartial reflectance and is related to the power coupling ratio into thecavity, κ_(k), by r_(k) ={square root}{square root over (1−κ _(k) )} andT is the feedback path round trip delay and is related to the freespectral range (FSR) by $T = {\frac{1}{FSR}.}$Any phase response can be engineered by varying the filter parametersφ_(k) and r_(k).

FIG. 6 shows an exemplary implementation of the architecture proposed inFIG. 1A using APF integrated on a planar light circuit. Output signalsfrom the fiber are split into two waveguides (waveguide 11 and waveguide12) by a polarization beam splitter 4. The polarization in waveguide 12is rotated by 90° using polarization rotator 6. Stage 1 is comprised ofa set of N₁ APF for each of the waveguides 11, 12 to generate phaseresponse of Φ_(1H)(ω) for waveguide 11 and Φ_(1V)(ω) for waveguide 12.In Stokes space, transmission through Stage 1 corresponds to a rotationabout {1,0,0} with rotation angle (Φ_(1V)(ω)−Φ_(1H)(ω)) Stage 2 iscomprised of a 50/50 directional coupler 8 with matched propagationconstants, followed by another set of N₂ APF for each of the waveguides11, 12, and then by another 50/50 directional coupler 10 with matchedpropagation constants.

This set of APF generates a phase response of Φ_(2H)(ω) for waveguide 11and Φ_(2V)(ω) for waveguide 12. In Stokes space, the first 50/50directional coupler gives a 90° rotation about {0,1,0}, the APF portionof Stage 2 is a rotation about {1,0,0} with rotation angle(Φ_(2V)(ω)−Φ_(2H)(ω)) and the second 50/50 directional coupler 10 isdesigned to give a 270° rotation about {0,1,0}. Thus, the combinedtransformation of Stage 2 is equivalent to a rotation about {0,0,1} withrotation angle of (Φ_(2V)(ω)−Φ_(2H)(ω)). Stage 3 is again another set ofN₃ APF for each of the waveguides 11, 12 to generate phase response ofΦ_(3H)(ω) for waveguide 11 and Φ_(3V)(ω) for waveguide 12. Transmissionthrough Stage 3 again corresponds to a rotation about {1,0,0} withrotation angle (Φ_(2V)(ω)−Φ_(2H)(ω)). After Stage 3, the twopolarizations are recombined on a single waveguide 12 via a polarizationrotator 14 and a polarization beam combiner 16. Stage 4 comprises ofanother set of APF to compensate the isotropic dispersion accumulatedthrough Stage 1 to 3.

After knowing the required rotation angles of θ₁(ω), θ₂(ω) and θ₃(ω) forall-frequency PMD compensation using equations (6-9) and (12-13), thecorresponding Φ_(1H)(ω) and Φ_(1V)(ω) can be derived fromΦ_(iV)(ω)−Φ_(iH)(ω)=θ_(i)(ω)   (15)One can arbitrarily chooseΦ_(iV)(ω)+Φ_(iH)(ω)=2N _(i)(π−ωT) for i=1,2,3   (16)so that the net isotropic dispersion introduced by the APF in bothwaveguides is simply a group delay. From equations (15) and (16), onegets $\begin{matrix}{{\Phi_{iV}(\omega)} = {{N_{i}( {\pi - {\omega\quad T}} )} + \frac{\theta_{i}(\omega)}{2}}} & (17)\end{matrix}$

To obtain the parameters φ_(k) and r_(k) for each APF in each waveguide,one fits equation (14) with equations (17) and (18) using a nonlinearfit subroutine that is available in standard Mathematical softwarepackages. With small N_(i), this nonlinear fit can be fast andefficient. Note that if the fitting is perfect, from equation (16), thefirst three stages only introduce isotropic dispersion$\frac{( {{\Phi_{iV}(\omega)} + {\Phi_{iH}(\omega)}} )}{2}$in the form of group delay. Thus, Stage 4 is redundant. However, inreality, since the fitting is done separately for equation (17) and(18), this isotropic dispersion often deviates from a simple groupdelay, and Stage 4 is needed for compensating these accumulatedisotropic dispersions.

To test the compensation scheme, two simulations are used. For bothsimulations, the mean DGD are chosen to be ˜70% of the bit period sothat higher order PMD dominates the signal's degradation. These meanDGDs are a factor of ˜7 larger than the tolerable first order PMD. Forthe first simulation, 500 fibers are randomly generated with mean DGD of17.5 ps by cascading 15 randomly oriented birefringence sections. Thebirefringence of these sections is Gaussian distributed with mean valueof 4.9 ps and with standard deviation of 20% of this mean value. Theinput signal is a 40 Gbit/s RZ pseudo-random bit sequences (2⁶-1) ofGaussian pulses of 10 ps (FHWM) pulse-width. For each generated fiber,one can compute the output PMD vector, {right arrow over (τ)}(ω), as afunction of frequency with optical frequency step-size of Δf=0.63 GHz.Based on this PMD vector data, one can compute the required rotationangles θ₁(ω), θ₂(ω) and θ₃(ω) using Eqs. (6-9) and (12-13).

From a randomly chosen fiber, the required rotation angles for thevarious stages are plotted as solid curves in FIGS. 3A-3C. FIG. 3A isfor θ₁(ω) of Stage 1, FIG. 3B is for θ₂(ω) of Stage 2, and FIG. 3C isfor θ₃(ω) of Stage 3. The thick solid curve in FIG. 3D shows the outputsignal from the fiber while the thin solid curve is the input signal.Using the exact profiles of the required rotation angles θ₁(ω), θ₂(ω)and θ₃(ω), the signal after passing through the PMD compensator is givenby the curve of unfilled triangles. The compensation is near perfect. Toillustrate that the synthesis of these rotation angles using APF isfeasible, one can fit Eqs. (17) and (18) using the “NonlinearFit”subroutine in Mathematica. The number of APF used are N₁=N₂=N₃=5. Thesynthesized rotation angles using the APF are plotted as dashed linecurve in FIGS. 3A-3C. The signal after PMD compensation using thesynthesized rotation angles from APF is shown as the curve of filledcircles in FIG. 3D. The signal is recovered to almost its initial shape.

FIGS. 4A-4D shows eye-diagrams of the signal by overlapping “ones” and“zeros” of the 500 randomly generated fiber. FIG. 4A is the eye-diagramof output signal from the fiber before compensation. The mean DGD of17.5 ps is so large that there is hardly any opening in the “eye”. FIG.4C shows the eye-diagram of the signal after the PMD compensator usingexact rotation angles calculated from Eqs. (6-9) and (12-13). The “eye”is recovered to near perfect. FIG. 4D shows the eye-diagram of thesignal after the PMD compensator using synthesized rotation angles fromAPF. It is evident that the quality of the signal has also improvedsignificantly. To illustrate that higher order PMD dominates thesignal's degradation when the mean DGD is large, one can pass the outputsignal from fiber into a first order PMD compensator and plot itseye-diagram in FIG. 4B. This first-order PMD compensator is comprised ofa polarization controller followed by a variable DGD. The “eye” hardlyopens up which confirms that higher order PMD significantly degrades thesignal.

For the second simulation, another 500 fibers are randomly generatedwith mean DGD of 4.35 ps by cascading 15 randomly oriented birefringencesections. The birefringence of these sections is Gaussian distributedwith mean value of 1.22 ps and with standard deviation of 20% of thismean value. This time, the input signal is a 160 Gbit/s RZ pseudo-randombit sequences (2⁶-1) of Gaussian pulses of 2.5 ps (FHWM) pulse-width.The step-size of the optical frequency Δf used is 2.54 GHz. Theabovementioned compensation procedures are again applied and theeye-diagrams are plotted in FIGS. 5A-5C.

FIG. 5A is shows the output signal from the fiber before compensation,and FIG. 5B shows the signal after the PMD compensator using exactrotation angles calculated from equations (6-9) and (12-13). FIG. 5Cshows the eye-diagram of the signal after the PMD compensator usingsynthesized rotation angles from APF. Significant improvement to thesignal's quality is again evident even for 160 Gbit/s transmission bitrate. With new generation low-PMD fiber, the mean DGD of the fiber canbe as low as 0.05 ps/≈{square root over (km)}. Thus, the mean DGD of4.35 ps used in the simulation corresponds to a propagation distance of˜7500 km. The encouraging results from the simulation seems to indicatethat PMD compensation schemes may help to make 160 Gbit/s transmissionthrough many thousands of kilometers feasible if new generation oflow-PMD fiber is deployed.

The inventive architecture of an All-Frequency PMD compensator in afeedforward compensation scheme is provided. It is comprised of 4stages. The first two stages give an equivalent frequency dependentpolarization rotation effect, the third stage provides the frequencydependent variable DGD while the last stage compensates for theisotropic dispersion created by the first three stages. In Stokes spaceformulation, the algorithm is described to find the required rotationangles of each stages using PMD concatenation rules. These rotationangles are shown can be implemented using APF in a compact integratedoptics circuit. Through simulations of 40 Gbits/s and 160 Gbit/stransmission, one can illustrate significant improvement of the signalquality using the compensator and also give evidence that 160 Gbit/stransmission through several thousand kilometres of propagation may befeasible with new generation of low-PMD fiber and the all-frequency PMDcompensator.

The invention also provides for an implementation of a deterministicAll-Frequency PMD (AFPMD) emulator that can generate any desired PMDvector for all frequencies. In the other words, one can “dial-in” anyspectrum of PMD vector that comprises of all orders of PMD. Thisemulator employs four stages as shown in FIG. 6A. At each angularfrequency ω (ω=2πf where f is the optical frequency), the “dial-in” PMDvector {right arrow over (τ)}_(e)(ω) has three parameters$\begin{pmatrix}{\tau_{ex}(\omega)} \\{\tau_{ey}(\omega)} \\{\tau_{ez}(\omega)}\end{pmatrix}.$The first three stages of the emulator provide the three degrees offreedom for emulating the PMD at each frequency. Stage 4 onlycompensates the isotropic dispersion introduced by the first threestages.

In Stokes space, Stage 1 and 3 are rotations about {1,0,0} with rotationangle θ₁(ω)and θ₃(ω) respectively. Stage 2 is a rotation about the{0,0,1} with rotation angle θ₂(ω). Rotation angles θ₁(ω), θ₂(ω) andθ₃(ω) are general functions of frequency, in contrast to the lineardependence on frequency of a birefringent element. The rotation matricesof Stages 1 and 3 are $\begin{matrix}{{R_{i}(\omega)} = {{\begin{pmatrix}1 & 0 & 0 \\0 & {\cos\quad{\theta_{i}(\omega)}} & {{- \sin}\quad{\theta_{i}(\omega)}} \\0 & {\sin\quad{\theta_{i}(\omega)}} & {\cos\quad{\theta_{i}(\omega)}}\end{pmatrix}\quad{for}\quad i} = {1\quad{and}\quad 3}}} & (19)\end{matrix}$and the rotation matrix of Stage 2 is $\begin{matrix}{{R_{2}(\omega)} = \begin{pmatrix}{\cos\quad{\theta_{2}(\omega)}} & {{- \sin}\quad{\theta_{2}(\omega)}} & 0 \\{\sin\quad{\theta_{2}(\omega)}} & {\cos\quad{\theta_{2}(\omega)}} & 0 \\0 & 0 & 1\end{pmatrix}} & (20)\end{matrix}$

Since${\overset{->}{\tau} \times \frac{\mathbb{d}R}{\mathbb{d}\omega}R^{+}},$the corresponding PMD vectors of Stage 1, 2, and 3 are${{{\overset{->}{\tau}}_{1}(\omega)} = \begin{pmatrix}\frac{\mathbb{d}{\theta_{1}(\omega)}}{\mathbb{d}\omega} \\0 \\0\end{pmatrix}},{{{\overset{->}{\tau}}_{2}(\omega)} = {{\begin{pmatrix}0 \\0 \\\frac{\mathbb{d}{\theta_{2}(\omega)}}{\mathbb{d}\omega}\end{pmatrix}\quad{and}\quad{{\overset{->}{\tau}}_{3}(\omega)}} = \begin{pmatrix}\frac{\mathbb{d}{\theta_{3}(\omega)}}{\mathbb{d}\omega} \\0 \\0\end{pmatrix}}}$respectively. To account for the input state of polarization (SOP), onecan transform the concatenated PMD to the input plane. With the PMDconcatenation rule, the PMD vector after Stage 3 transformed to theinput plane of the emulator is given as $\begin{matrix}\begin{matrix}{{{\overset{->}{\Gamma}}_{3s}(\omega)} = {{{\overset{->}{\tau}}_{1}(\omega)} + {{R_{1}^{+}(\omega)}{{\overset{->}{\tau}}_{2}(\omega)}} + {{R_{1}^{+}(\omega)}{R_{2}^{+}(\omega)}{{\overset{->}{\tau}}_{3}(\omega)}}}} \\{= \begin{pmatrix}{\frac{\mathbb{d}\theta_{1}}{\mathbb{d}\omega} + {\cos\quad\theta_{2}\frac{\mathbb{d}\theta_{3}}{\mathbb{d}\omega}}} \\{{\sin\quad\theta_{1}\frac{\mathbb{d}\theta_{2}}{\mathbb{d}\omega}} - {\cos\quad\theta_{1}\sin\quad\theta_{2}\frac{\mathbb{d}\theta_{3}}{\mathbb{d}\omega}}} \\{{\cos\quad\theta_{1}\frac{\mathbb{d}\theta_{2}}{\mathbb{d}\omega}} + {\sin\quad\theta_{1}\sin\quad\theta_{2}\frac{\mathbb{d}\theta_{3}}{\mathbb{d}\omega}}}\end{pmatrix}}\end{matrix} & (21)\end{matrix}$For a given “dial-in” PMD vector spectrum${{{\overset{->}{\tau}}_{e}(\omega)} = \begin{pmatrix}{\tau_{ex}(\omega)} \\{\tau_{ey}(\omega)} \\{\tau_{ez}(\omega)}\end{pmatrix}},$one can set {right arrow over (Γ)}_(3s)(ω)={right arrow over (τ)}_(e)(ω)and solve to obtain $\begin{matrix}{{\frac{\mathbb{d}\theta_{1}}{\mathbb{d}\omega}(\omega)} = {{\cot\quad{{\theta_{2}(\omega)}\lbrack {{{\tau_{ey}(\omega)}\cos\quad{\theta_{1}(\omega)}} - {{\tau_{ez}(\omega)}\sin\quad{\theta_{1}(\omega)}}} \rbrack}} + {\tau_{ex}(\omega)}}} & (22) \\{{\frac{\mathbb{d}\theta_{2}}{\mathbb{d}\omega}(\omega)} = {{{\tau_{ey}(\omega)}\sin\quad{\theta_{1}(\omega)}} + {{\tau_{ez}(\omega)}\cos\quad{\theta_{1}(\omega)}}}} & (23) \\{{\frac{\mathbb{d}\theta_{3}}{\mathbb{d}\omega}(\omega)} = {\csc\quad{{\theta_{2}(\omega)}\lbrack {{{\tau_{ez}(\omega)}\sin\quad{\theta_{1}(\omega)}} - {{\tau_{ey}(\omega)}\cos\quad{\theta_{1}(\omega)}}} \rbrack}}} & (24)\end{matrix}$Note that this “dial-in” PMD vector spectrum {right arrow over(τ)}_(e)(ω) is for the input plane. The synthesis algorithm for therequired rotation angles of the three stages is as followed: At aninitial frequency, ω_(initial), one can arbitrarily fix the rotationangles of Stage 1, 2 and 3 to be 0, π/2 and 0 respectively. Theseθ₁(ω_(initial)), θ₂(ω_(initial)) and θ₃(ω_(initial)) provide thestarting points for the algorithm. For subsequent frequencies, therotation angles are found in a step-wise manner: $\begin{matrix}{{{{\theta_{i}( {\omega + {\Delta\quad\omega}} )} \approx {{\theta_{i}(\omega)} + {\frac{\mathbb{d}\theta_{i}}{\mathbb{d}\omega}(\omega)\Delta\quad\omega\quad{for}\quad i}}} = 1},{2\quad{and}\quad 3}} & (25)\end{matrix}$where the $\frac{\mathbb{d}\theta_{i}}{\mathbb{d}\omega}(\omega)$are given by equations (22-28). In this way, one can successivelysynthesize the required profiles of the rotation angles of the firstthree stages to generate any desired spectrum of PMD vectors {rightarrow over (τ)}_(e)(ω).

One promising implementation of the proposed architecture is shown inFIG. 6B. It is based on All-Pass Filters (APFs) integrated on a planarlightwave circuit. However, the required phase responses of thesefilters are distinctly different from other approaches to approximatethe inverse Jones matrix while the aim is to emulate itsfrequency-dependent part (i.e. jU⁺U_(ω)(ω)). The incoming signals aresplit into two waveguides (waveguide 50 and waveguide 52) by apolarization beam splitter. The polarization in waveguide 52 is rotatedby 90°. Stage 1 is comprised of a set of N₁ APFs for each of thewaveguides to generate phase responses of Φ_(1H)(ω) for waveguide 50 andΦ_(1V)(ω) for waveguide 52. In Stokes space, transmission through Stage1 corresponds to a rotation about {1,0,0} with rotation angle(Φ_(1V)(ω)−Φ_(1H)(ω).

Stage 2 is comprised of a 50/50 directional coupler 58 with matchedpropagation constants, followed by another set of N₂ APFs for each ofthe waveguides 50, 52, and then by another 50/50 directional coupler 60with matched propagation constants. This set of APFs generates phaseresponses of Φ_(2H)(ω) for waveguide 50 and Φ_(2V)(ω) for waveguide 52.In Stokes space, the first 50/50 directional coupler 58 gives a 90°rotation about {0,1,0}, the APF portion of Stage 2 is a rotation about{1,0,0} with rotation angle (Φ_(2V)(ω)−Φ_(2H)(ω)) and the second 50/50directional coupler 60 is designed to give a 270  rotation about{0,1,0}. Thus, the combined transformation of Stage 2 is equivalent to arotation about {0,0,1} with rotation angle of (Φ_(2V)(ω)−Φ_(2H)(ω)).Stage 3 is again another set of N₃ APFs for each of the waveguides 50,53 to generate a phase response of Φ_(3H)(ω) for waveguide 50 andΦ_(3V)(ω) for waveguide 52.

Transmission through Stage 3 again corresponds to a rotation about{1,0,0} with rotation angle (Φ_(3V)(ω)−Φ_(3H)(ω)). After Stage 3, thetwo polarizations are recombined on a single waveguide 52 via apolarization rotator 62 and a polarization beam combiner 64. Stage 4comprises of another set of APFs to compensate the isotropic dispersionaccumulated through Stage 1 to 3.

All-pass filters (APFs) have unity magnitude response; and, by cascadingthem, one can engineer the phase response to approximate any desiredresponse. Two common implementations of optical APF are a) a ringresonator coupled to a straight waveguide and b) an etalon that has aperfect mirror on one side. The former is preferred for compactintegrated optics. The latter is also known as a Gires-Toumoisinterferometer. The phase response of a set of N_(i) APFs is given by:$\begin{matrix}{{\Phi_{i}(\omega)} = {{N_{i}( {\pi - {\omega\quad T}} )} - \quad{\sum\limits_{k}^{N_{i}}\quad( {\phi_{k} + {2\quad{\tan^{- 1}( \frac{r_{k}\quad{\sin( {{\omega\quad T} + \phi_{k}} )}}{1 - {r_{k}\quad{\cos( {{\omega\quad T} + \phi_{k}} )}}} )}}} )}}} & (26)\end{matrix}$where φ_(k) determines the cavity's resonant frequency, r_(k) is thepartial reflectance and is related to the power coupling ratio into thecavity, κ_(k), by r_(k) =≈{square root over (1−κ _(k) )} and T is thefeedback path round trip delay and is related to the free spectral range(FSR) by $T = {\frac{1}{FSR}.}$Any phase response can be engineered by varying the filter parametersφ_(k) and r_(k).

After determining the required rotation angles θ₁(ω), θ₂(ω) and θ₃(ω)using equations (22-25), the corresponding Φ_(iH)(ω) and Φ_(iV)(ω) canbe derived fromΦ_(iV)(ω)−Φ_(iH)(ω)=θ₁(ω)   (27)One can arbitrarily chooseΦ_(iV)(ω)+Φ_(iH)(ω)=2N _(i)(π−ωT) for i=1,2 and 3   (28)so that the net isotropic dispersion introduced by the APF in bothwaveguides is simply a group delay. From equations (9) and (10), one canwrite $\begin{matrix}{{\Phi_{iV}(\omega)} = {{N_{i}\quad( {\pi - {\omega\quad T}} )} + \frac{\theta_{i}(\omega)}{2}}} & (29) \\{{\Phi_{iH}(\omega)} = {{N_{i}( {\pi - {\omega\quad T}} )} - \frac{\theta_{i}(\omega)}{2}}} & (30)\end{matrix}$

To obtain the parameters φ_(k) and r_(k) for each APF in each waveguide,one can fit equation (26) with equations (29) and (30) using a nonlinearfit subroutine that is available in standard mathematics softwarepackages. With small N_(i), this nonlinear fit can be fast andefficient. Note that if the fitting is perfect, from equation (29), thefirst three stages only introduce isotropic dispersion$\frac{( {{\Phi_{iV}(\omega)} + {\Phi_{iH}(\omega)}} )}{2}$in the form of a group delay. Thus, Stage 4 is redundant. However, inreality, since the fitting is done separately for equation (29) and(30), this isotropic dispersion often deviates from a simple groupdelay, and Stage 4 is needed to compensate the accumulated isotropicdispersion.

To test the AFPMD emulator, one can randomly simulate fibers bycascading 30 randomly oriented birefringence sections. The birefringenceof these sections is Gaussian distributed with a mean value of 0.86 psand with a standard deviation of 20% of this mean value. Thiscorresponds to a mean differential group delay of 4.35 ps. Theconcatenated PMD vectors of this simulated fiber of 30 sections arecomputed over a frequency range of ±250 GHz about the carrier's opticalfrequency f_(o) with a step-size Δf=2.54 GHz. The PMD vectors aretransformed to the input plane. This spectrum serves as the “dial-in”PMD vector spectrum for the AFPMD emulator.

Based on this spectrum, the required rotation angles θ₁(ω), θ₂(ω) andθ₃(ω) are computed using equation (22-25) and then fit equations (29)and (30) using the “NonlinearFit” subroutine in Mathematica. The numberof APFs used are N₁=N₂=N₃=3. All APF have the same FSR of 500 GHz. FIG.7 shows the rotational angles for a simulated fiber randomly chosen fromthe ensemble. The solid line curves are the required rotation angleswhile the curves of crosses are the rotation angles synthesized by theAPF.

To investigate the performance of the AFPMD emulator, one can send atrain of 160 Gbit/s RZ pseudo-random bit sequences (2⁶-1) of Gaussianpulses of 2.5 ps (FHWM) pulse-width through this particular simulatedfiber of 30 sections and also through the AFPMD emulator with the sameinput SOP. If their output signals match, it indicates that the AFPMDemulator has produced the same PMD spectrum as that of the simulatedfiber of 30 sections. FIG. 8 shows both the output signals. The solidline curve is for the simulated fiber of 30 sections while the curve offilled circles is for the AFPMD emulator. The input signal is shown asthe dashed line. The two output signals match closely, therebydemonstrating the capability of the AFPMD emulator in generating anarbitrary PMD spectrum. Good agreements are also observed for otherfiber realizations in the simulations.

The architecture of an All-Frequency PMD emulator based on four stagesof APF is provided for by the invention. The first three stages providethe three degrees of freedom for emulating PMD at each frequency, whilethe last stage compensates for the isotropic dispersion created by thefirst three stages. In Stokes space formulation, one can describe thealgorithm to synthesize the required rotation angle of each stage usingPMD concatenation rules. Also, these rotation angles can be implementedusing APFs in a compact integrated optics circuit. Through simulationsof 160 Gbit/s transmission, one can demonstrate good fidelity of theAFPMD emulator in generating any desired spectrum of PMD vectors.

Although the present invention has been shown and described with respectto several preferred embodiments thereof, various changes, omissions andadditions to the form and detail thereof, may be made therein, withoutdeparting from the spirit and scope of the invention.

1. A polarization mode dispersion (PMD) device comprising: a first stagethat receives a signal and performs rotation about {1,0,0} on saidsignal, said first stage outputs a first signal; a second stage thatreceives said first signal and performs a rotation about {0,0,1} on saidfirst signal, said second stage outputs a second signal that representsthe alignment of the PMD of various frequencies into a commondirection;, and a third stage that receives said second signal andprovides the necessary frequency dependent variable differential delaygroup (DGD) in the {1,0,0} direction to cancel the PMD in any specifiedfrequency range.
 2. The PMD device of claim 1 further comprising afourth stage that compensates for the isotropic dispersion introduced bysaid first, second and third stages due to imperfect fitting of rotationangles.
 3. The PMD device of claim 1, wherein said first stage comprisesa rotation angle θ₁(ω) as a function of frequency.
 4. The PMD device ofclaim 3, wherein said first stage comprises a rotation angle θ₂(ω) as afunction of frequency.
 5. The PMD device of claim 1, wherein said firststage comprises all pass filters (APF) or any component that cangenerate flexible frequency-dependent phase profiles.
 6. The PMD deviceof claim 1, wherein said second stage comprises all pass filters (APF)or any component that can generate flexible frequency-dependent phaseprofiles.
 7. The PMD device of claim 1, wherein said third stagecomprises all pass filters (APF) or any component that can generateflexible frequency-dependent phase profiles.
 8. The PMD device of claim1, wherein said second stage comprises a 50/50 directional coupler withmatched propagation constants.
 9. The PMD device of claim 4, whereinthird stage comprises a rotation angle θ₃(ω) as a function of frequency.10. The PMD device of claim 9, wherein said third stage comprises thefollowing relation to establish compensator${\theta_{3}( {\omega + {\Delta\quad\omega}} )} \approx {{\theta_{3}(\omega)} + {\frac{\mathbb{d}\theta_{3}}{\mathbb{d}\omega}(\omega)\quad\Delta\quad{\omega.}}}$11. The PMD device of claim 9, wherein said first, second, and thirdstages comprise a relation, $\begin{matrix}{{\theta_{i}( {\omega + {\Delta\quad\omega}} )} \approx {{\theta_{i}(\omega)} + {\frac{\mathbb{d}\theta_{i}}{\mathbb{d}\omega}(\omega)\quad\Delta\quad\omega}}} & {{{{for}\quad i} = 1},2,{{and}\quad 3},}\end{matrix}$ that defines a compensator.
 12. A method of operating apolarization mode dispersion (PMD) device comprising: receiving at afirst stage a signal and performing rotation about {1,0,0} on saidsignal, said first stage outputs a first signal; receiving at a secondstage said first signal and performs a rotation about {0,0,1} on saidfirst signal, said second stage outputs a second signal that representsthe alignment of the various PMD frequencies into a common direction;and receiving at a third stage said second signal and provides thenecessary frequency dependent variable differential group delay (DGD) inthe {1,0,0} direction to cancel the PMD in any specified frequencyrange.
 13. The method of claim 12 further comprising compensating at afourth stage the isotropic dispersion introduced by said first, secondand third stages due to imperfect fitting of rotation angles.
 14. Themethod of claim 12, wherein said first stage comprises a rotation angleθ₁(ω) as a function of frequency.
 15. The method of claim 14, whereinsaid first stage comprises a rotation angle 02(6)) as a function offrequency.
 16. The method of claim 12, wherein said first stagecomprises all pass filters (APF) or any component that can generateflexible frequency-dependent phase profiles.
 17. The method of claim 12,wherein said second stage comprises all pass filters (APF) or anycomponent that can generate flexible frequency-dependent phase profiles.18. The method of claim 12, wherein said third stage comprises all passfilters (APF) or any component that can generate flexiblefrequency-dependent phase profiles.
 19. The PMD device of claim 12,wherein said second stage comprises a 50/50 directional coupler withmatched propagation constants.
 20. The method of claim 12, wherein thirdstage comprises a rotation angle θ₃(ω) as a function of frequency. 21.The method of claim 20, wherein said third stage comprises the followingrelation to establish compensator${\theta_{3}( {\omega + {\Delta\quad\omega}} )} \approx {{\theta_{3}(\omega)} + {\frac{\mathbb{d}\theta_{3}}{\mathbb{d}\omega}(\omega)\quad\Delta\quad{\omega.}}}$22. The method of claim 20, wherein said first, second, and third stagescomprise a relation, $\begin{matrix}{{\theta_{i}( {\omega + {\Delta\quad\omega}} )} \approx {{\theta_{i}(\omega)} + {\frac{\mathbb{d}\theta_{i}}{\mathbb{d}\omega}(\omega)\quad\Delta\quad\omega}}} & {{{{for}\quad i} = 1},2,{{and}\quad 3},}\end{matrix}$ that defines a compensator.
 23. A polarization modedispersion (PMD) compensator comprising: a first stage that receives asignal and performs rotation about {1,0,0} on said signal, said firststage outputs a first signal; a second stage that receives said firstsignal and performs a rotation about {0,0,1} on said first signal, saidsecond stage outputs a second signal that represents the alignment ofthe various PMD frequencies into a common direction; and a third stagethat receives said second signal and provides the necessary frequencydependent variable in the {1,0,0} direction to cancel the PMD in anyspecified frequency range.
 24. The PMD compensator of claim 23 furthercomprising a fourth stage that compensates for the isotropic dispersionintroduced by said first, second and third stages due to imperfectfitting of rotation angles.
 25. The PMD compensator of claim 23, whereinsaid first stage comprises a rotation angle θ₁(ω) as a function offrequency.
 26. The PMD compensator of claim 25, wherein said first stagecomprises a rotation angle θ₂(ω) as a function of frequency.
 27. The PMDcompensator of claim 23, wherein said first stage comprises all passfilters (APF) or any component that can generate flexiblefrequency-dependent phase profiles.
 28. The PMD compensator of claim 23,wherein said second stage comprises all pass filters (APF) or anycomponent that can generate flexible frequency-dependent phase profiles.29. The PMD compensator of claim 23, wherein said third stage comprisesall pass filters (APF) or any component that can generate flexiblefrequency-dependent phase profiles.
 30. The PMD compensator of claim 23,wherein said second stage comprises a 50/50 directional coupler withmatched propagation constants.
 31. The PMD compensator of claim 26,wherein third stage comprises a rotation angle θ₃(ω) as a function offrequency.
 32. The PMD compensator of claim 31, wherein said third stagecomprises the following relation to establish compensator${\theta_{3}( {\omega + {\Delta\quad\omega}} )} \approx {{\theta_{3}(\omega)} + {\frac{\mathbb{d}\theta_{3}}{\mathbb{d}\omega}(\omega)\quad\Delta\quad{\omega.}}}$33. A polarization mode dispersion (PMD) emulator comprising: a firststage that receives a signal and performs rotation about {1,0,0} on saidsignal, said first stage outputs a first signal; a second stage thatreceives said first signal and performs a rotation about {0,0,1} on saidfirst signal, said second stage outputs a second signal that representsthe alignment of the various PMD frequencies into a common direction;and a third stage that receives said second signal and provides thenecessary frequency dependent variable in the {1,0,0} direction tocancel the PMD in any specified frequency range.
 34. The PMD emulator ofclaim 33 further comprising a fourth stage that compensates for theisotropic dispersion introduced by said first, second and third stagesdue to imperfect fitting of rotation angles.
 35. The PMD emulator ofclaim 33, wherein said first stage comprises a rotation angle θ₁(ω) as afunction of frequency.
 36. The PMD emulator of claim 35, wherein saidfirst stage comprises a rotation angle θ₂(ω) as a function of frequency.37. The PMD emulator of claim 33, wherein said first stage comprises allpass filters (APF) or any component that can generate flexiblefrequency-dependent phase profiles.
 38. The PMD emulator of claim 33,wherein said second stage comprises all pass filters (APF) or anycomponent that can generate flexible frequency-dependent phase profiles.39. The PMD emulator of claim 33, wherein said third stage comprises allpass filters or any component that can generate flexiblefrequency-dependent phase profiles (APF).
 40. The PMD emulator of claim33, wherein said second stage comprises a 50/50 directional coupler withmatched propagation constants.
 41. The PMD emulator of claim 36, whereinthird stage comprises a rotation angle θ₃(ω) as a function of frequency.42. The PMD emulator of claim 41, wherein said first, second, and thirdstages comprise a relation,${{{\theta_{i}( {\omega + {\Delta\quad\omega}} )} \approx {{\theta_{i}(\omega)} + {\frac{\mathbb{d}\theta_{i}}{\mathbb{d}\omega}(\omega)\quad\Delta\quad\omega\quad{for}\quad i}}} = 1},2,{{and}\quad 3},$that defines an emulator.